The quote within the quote in your Question is Book 5, Lines 255-258 of Ovid's Fasti. The full story runs from Line 229 through Line 260 of the same book, making up a medium-sized paragraph. (James George Frazer's translation thereof, which might make for less obscure reading than the one in the Kersey Graves book, can be found on The Theoi Project. A.S. Kline's translation preserves the verse format of the original but I also found it to be quite readable, available on Poetry in Translation.)
In it, the Fasti's narrator is having a conversation with Flora, the goddess of flowers, who is explaining to him different ways in which she is important to the land's fertility and thus the food and water sources of all living things. In the paragraph in question she points out how she is the reason that Mars, an important agricultural deity, was born in the first place.
The goddess Juno was deeply chagrined that her husband Jupiter was able to "give birth" to Minerva apparently without the usual means of needing female intervention for the feat. While on her way to complain about this to Oceanus, she made a pit-stop at Flora's house and opened up to the flower-deity about her grievance, revealing also that she had decided not to be outdone by her own spouse in the matter of childbearing. Determined to engender offspring "'without contact with a man'", she would "'try all the drugs in the wide world, and I will explore the seas and the depths of Tartarus.'"
Presumably because Juno was aware of Flora's deep fertility powers, she was convinced that the flower-goddess could help her with this goal. Flora was very hesitant because of the potential of incurring Jupiter's wrath but Juno persuaded her by swearing that Flora would never be implicated in this action.
At this, Flora told Juno of a certain one-of-a-kind flower in her garden "'that was sent [to] me from the fields of Olenus.'" Flora had been given testing instructions regarding the flower by "'He who gave it me'". Tantalisingly, she never says who "He" was, but whoever it was told her to touch a barren heifer with the flower and it would become a mother, and it worked as predicted.
Apparently in a ritualistic gesture, Flora now plucked the flower using her thumb, touched Juno's bosom with it, and the queen of the gods instantly became pregnant. Juno then journeyed to Thrace where she gave birth to Mars somewhere near "'the shores of the Propontis.'" The story ends with Mars, who evidently knows of Flora's involvement in his conception, gratefully granting her a place of honour in the Roman pantheon once the city of Rome is founded by his son Romulus.
Wikipedia's summary on this is that:
Ovid is the only source for the story. He may be presenting a literary
myth of his own invention, or an otherwise unknown archaic Italic
tradition; either way, in choosing to include the story, he emphasizes
that Mars was connected to plant life and was not alienated from
female nurture.
After citing this portion of the Fasti on The Theoi Project, Aaron Atsma makes a note that:
The fact that Ovid mentions the Greek city of Olenos [Olenus] in this myth,
strongly suggests it was derived it from a Greek source.
Whatever the validity of that position is, Mars having been born in Thrace is supposed to explain the warlike character of the tribes of that region, in which country Statius' Thebaid says the god also lived. Thebaid 4 suggests either that Mars was born during winter at a place called Odrysus in Thrace, or at least that he was there, crawling in the snow, at some point during his babyhood.
